Quadratic Formula and Completing the Square

A quadratic equation is an equation that can be written in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$. While some quadratic equations can be solved by factoring, not all of them can. For those that cannot be easily factored, we use two powerful methods: completing the square and the quadratic formula.

Completing the Square

Completing the square is a method used to rewrite a quadratic equation so that it contains a perfect square trinomial. This allows you to solve for $x$ by taking the square root of both sides.

Steps to complete the square for $x^2 + bx + c = 0$:

1. Move the constant term $c$ to the right side of the equation.
2. Take half of the $x$ coefficient ($b$), square it: $(\frac{b}{2})^2$.
3. Add this value to both sides of the equation.
4. Rewrite the left side as a squared binomial: $(x + \frac{b}{2})^2$.
5. Take the square root of both sides (remembering the $\pm$ symbol) and solve for $x$.

Example: Solve $x^2 + 6x + 2 = 0$

1. Move the constant to the right side: $x^2 + 6x = -2$
2. Take half of $6$ (which is $3$) and square it to get $9$. Add $9$ to both sides: $x^2 + 6x + 9 = -2 + 9$
3. Factor the left side into a perfect square: $(x + 3)^2 = 7$
4. Take the square root of both sides: $x + 3 = \pm\sqrt{7}$
5. Solve for $x$: $x = -3 \pm \sqrt{7}$

The Quadratic Formula

The quadratic formula is derived from completing the square and provides a direct way to find the solutions to any quadratic equation $ax^2 + bx + c = 0$.

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solve $3x^2 - 2x - 5 = 0$

Here, $a = 3$, $b = -2$, and $c = -5$. Plug these values into the formula:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-5)}}{2(3)}$ $x = \frac{2 \pm \sqrt{4 + 60}}{6}$ $x = \frac{2 \pm \sqrt{64}}{6}$ $x = \frac{2 \pm 8}{6}$

Now, calculate the two possible values for $x$:

• $x = \frac{2 + 8}{6} = \frac{10}{6} = \frac{5}{3}$
• $x = \frac{2 - 8}{6} = \frac{-6}{6} = -1$

The solutions are $x = \frac{5}{3}$ and $x = -1$.

The Discriminant

The expression inside the square root of the quadratic formula, $b^2 - 4ac$, is called the discriminant. It tells you exactly how many real solutions the quadratic equation has, without having to solve the entire equation.

• If $b^2 - 4ac > 0$: The equation has two distinct real solutions.
• If $b^2 - 4ac = 0$: The equation has exactly one real solution (a repeated root).
• If $b^2 - 4ac < 0$: The equation has no real solutions (the solutions are complex numbers).